Page:The principle of relativity (1920).djvu/143

 the increment of the proper time for a complete description of the orbit, finally nΤ = 2π, so that from a properly chosen initial point τ, we have the Kepler-equation

(29) nτ = E - e sin E.

If we now change the unit of time, and denote the velocity of light by c, then from (28), we obtain

(30) (dt/dτ)^2 - 1 = (m^ ac^2) (1 + e cos E)/(1 - e cos E)

Now neglecting c^{-4} with regard to 1, it follows that

ndt = ndτ [ 1 + 1/2 m^ ac^2 (1 + e cos E)/(1 - e cos E) ]

from which, by applying (29),

(31) nt + const = (1 + 1/2 m^ ac^2) nτ + m^ ac^2 Sin E.

the factor m^ ac^2 is here the square of the ratio of a certain average velocity of F in its orbit to the velocity of light. If now m^* denote the mass of the sun, a the semi major axis of the earth's orbit, then this factor amounts to 10^{-8}.

The law of mass attraction which has been just described and which is formulated in accordance with the relativity postulate would signify that gravitation is propagated with the velocity of light. In view of the fact that the periodic terms in (31) are very small, it is not possible to decide out of astronomical observations between such a law (with the modified mechanics proposed above) and the Newtonian law of attraction with Newtonian mechanics.